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Find largest number not representable by ax+by

Source:

September 29, 2010

number theoryDivisibilityFrobeniusAdditive Number TheoryIMO ShortlistIMO Longlist

Problem Statement

(GBR 2)

Let

a, b, x, y

be positive integers such that

a

and

b

have no common divisor greater than

1

. Prove that the largest number not expressible in the form

ax + by

ab - a - b

. If

N(k)

is the largest number not expressible in the form

ax + by

in only

k

ways, find

N(k).